I am reading David Skinner's notes on AQFT. In Chapter 1 page 3, he mentioned that "purely from the point of view of functions on $C$, locality on $M$ is actually a very strong restriction", with the following justification:
"Even a monomial function on $C$ generically looks like \begin{equation} \int_{M^{\otimes n}} \mathrm{~d}^d x_1 \mathrm{~d}^d x_2 \cdots \mathrm{d}^d x_n \Lambda\left(x_1, x_2, \ldots, x_n\right) \phi\left(x_1\right) \phi\left(x_2\right) \cdots \phi\left(x_n\right) \end{equation} involving the integral of the field at many different points, with some choice of function $\Lambda: M^{\otimes n} \rightarrow \mathbb{R}$. Locality means that we restrict to 'functions' $\Lambda$ of the form \begin{equation} \Lambda\left(x_1, x_2, \ldots, x_n\right)=\lambda(x) \partial^{\left(p_1\right)} \delta^d\left(x_1-x_2\right) \partial^{\left(p_2\right)} \delta^d\left(x_2-x_3\right) \cdots \partial^{\left(p_{n-1}\right)} \delta^d\left(x_{n-1}-x_n\right)\label{Eq:1} \end{equation} that are supported on the main diagonal $M \subset M^{\otimes n}$, with finitely many derivatives allowed to act on the $\delta$-functions. Integrating by parts if necessary, these derivatives can be made to act on the fields, leaving us with an expression of the general form \begin{equation} \int_M \mathrm{~d}^d x \lambda(x) \partial^{\left(q_1\right)} \phi(x) \partial^{\left(q_2\right)} \phi(x) \cdots \partial^{\left(q_n\right)} \phi(x) \label{2}\end{equation} of a monomial of degree $n$ in the fields, acted on again by some derivatives, with all fields and derivatives evaluated at the same $x \in M$. The only remnant of the original $\Lambda: M^{\otimes n} \rightarrow \mathbb{R}$ is the function $\lambda: M \rightarrow \mathbb{R}$. "
The discussion here looks a bit mysterious to me, could someone explain what he is saying essentially here?
Specifically, I wonder 1.how "locality constraints" imply the first equation? 2. What is locality in this case specifically meant? I presume locality in this case it's just requiring the Lagrangian density to be a function of field at localized points of spacetime? e.And how indeed is the second equation derived?
Also, I seldom see QFT textbooks in this highly geometrically inclined fashion, could you recommend me some of the similar books or notes?
